A Note on Generalized Erdős-Rogers Problems

Abstract: For a $k$-uniform hypergraph $F$ and positive integers $s$ and $N$, the generalized Erdős-Rogers function $f^{(k)}_{F,s}(N)$ denotes the largest integer $m$ such that every $K_s^{(k)}$-free $k$-graph on $N$ vertices contains an $F$-free induced subgraph on $m$ vertices. In particular, if $F = K^{(k)}_t$, then we write $f^{(k)}_{t,s}(N)$ for $f^{(k)}_{F,s}(N)$. Mubayi and Suk (\emph{J. London. Math. Soc. 2018}) conjectured that $f^{(4)}_{5,6}(N)=(\log \log N)^{Θ(1)}$. Motivated by this conjecture, we prove that $f^{(4)}_{5^{-},6}(N)=(\log\log N)^{Θ(1)}$, where $5^{-}$ denotes the $4$-graph obtained from $K_5^{(4)}$ by deleting one edge. Our proof combines a probabilistic construction of a $2$-coloring of pairs with a stepping-up construction and an analysis of multi-layer local extremum structures. Furthermore, we derive an upper bound for a more general Erdős-Rogers function, which implies the lower bound $r_4(6,n)\ge 2^{2^{cn^{1/2}}}$. By applying a variant of the Erdős-Hajnal stepping-up lemma due to Mubayi and Suk, we also slightly improve the lower bound for $r_k(k+2,n)$.

• The paper establishes that the generalized Erdős-Rogers function f^(4)_{5^-,6}(N) grows as (log log N)^(Θ(1)), settling a relaxed Mubayi–Suk conjecture.
• It employs advanced probabilistic constructions, refined stepping-up lemmas, and multi-layer sequence analysis to navigate the challenges of forbidden subgraph configurations.
• The techniques yield improved lower bounds for off-diagonal hypergraph Ramsey numbers, offering insights for both extremal combinatorics and complexity theory.

Summary of "A Note on Generalized Erdős-Rogers Problems" (2604.02835)

Introduction and Context

The paper investigates the generalized Erdős-Rogers function $f^{(k)}_{F,s}(N)$ in the context of $k$-uniform hypergraphs, focusing on induced subgraph avoidance properties in $K_s^{(k)}$-free $k$-graphs. Specifically, the authors address a conjecture by Mubayi and Suk related to the precise growth order of $f^{(4)}_{5,6}(N)$, i.e., the largest order of a $K_5^{(4)}$-free induced subgraph contained in every $K_6^{(4)}$-free $4$-graph on $N$ vertices. This question is tightly connected to classic and off-diagonal hypergraph Ramsey theory, a topic of persistent difficulty and central importance in extremal combinatorics.

Main Results and Techniques

The central achievement is the determination of the correct order of growth for

$f^{(4)}_{5^-,6}(N) = (\log \log N)^{\Theta(1)},$

where $k$0 denotes the $k$1-uniform hypergraph obtained from $k$2 by deleting an edge. The result matches the predicted growth for $k$3 but for this slightly weakened forbidden subhypergraph, and thereby essentially settles the Mubayi–Suk conjecture for a relaxed case.

The methodology incorporates several sophisticated combinatorial and probabilistic techniques:

• Probabilistic Construction of Colorings: The deployment of random colorings to guarantee, with high probability, that any large enough subset contains a configuration critical to the embedding or avoidance problem.
• Erdős–Hajnal Stepping-Up Lemma and Variants: The classic lemma is employed and refined to transfer (lower or upper) bound constructions from lower-dimensional graphs to higher-uniformity hypergraphs.
• Analysis of Local Extremum Structures: Detailed analysis of local maxima and monotonicity in binary representations guides the construction and avoidance of forbidden configurations, particularly in the "stepping-up" process.
• Multi-layer Sequence Analysis: The greedy selection and hierarchical analysis of sequences of local maxima or monotone sequences underpin the independence and avoidance arguments for the induced subgraphs.

A key component is the construction of a $k$4-free $k$5-graph $k$6 such that $k$7, confirming that even for large $k$8, every $k$9-free $K_s^{(k)}$0-graph necessarily contains only relatively small $K_s^{(k)}$1-free induced subgraphs. Thus, the double-logarithmic bound is tight up to constants in the exponent.

Corollaries and Improved Bounds

By leveraging a variant of the stepping-up lemma, the authors extend their result to higher uniformities:

$K_s^{(k)}$2

for every $K_s^{(k)}$3, where $K_s^{(k)}$4 is the $K_s^{(k)}$5-graph on $K_s^{(k)}$6 vertices with four edges. This general principle highlights a broad phenomenon concerning the minimum size of induced $K_s^{(k)}$7-free subgraphs within $K_s^{(k)}$8-free $K_s^{(k)}$9-graphs across dimensions.

The paper also improves the lower bounds for certain off-diagonal hypergraph Ramsey numbers. Specifically,

$k$0

for some absolute constant $k$1, pushes the best known exponent higher, and the stepping-up approach propagates these improvements to higher-uniformity Ramsey numbers:

$k$2

for $k$3, where $k$4 is the $k$5-fold tower function.

Contradictory Claims and Open Problems

• Sharpness of the Upper Bound: The established upper bound for $k$6 matches the lower bound up to constants, confirming the predicted double-logarithmic growth.
• Boundary of the Stepping-Up Approach: The last open case related to the classical tower height for off-diagonal Ramsey numbers is $k$7; a positive resolution would imply the longstanding conjecture that $k$8 is double-exponential in $k$9.
• Limitation for Further Generalization: The approach confirms sharp results for $f^{(4)}_{5,6}(N)$0 but leaves the key case $f^{(4)}_{5,6}(N)$1 open, emphasizing the subtlety at the boundary between extremal and probabilistic combinatorics in hypergraph Ramsey theory.

Implications and Prospects

Theoretical Implications

These results:

• Clarify the fine structure of generalized Erdős-Rogers functions for hypergraphs close to the cluster point of clique numbers, realizing conjectured bounds in a challenging setting.
• Demonstrate the power and limits of stepping-up and probabilistic methods in transferring and amplifying Ramsey-type lower bounds across hypergraph uniformities.
• Suggest that double-logarithmic (and, more generally, iterated logarithmic) phenomena naturally demarcate the rapid boundary transitions in forbidden induced subgraph problems for hypergraphs.

Practical Implications

While the work lies squarely in extremal combinatorics, advances in hypergraph Ramsey numbers have downstream relevance in random structure theory, property testing, theoretical computer science (e.g., lower bounds in communication, circuit complexity), and the study of phase transitions in random hypergraphs.

Future Developments

Refinement of Stepping-Up Techniques: Enhanced variants of the stepping-up lemma tailored for specific forbidden configurations could potentially close the remaining gaps for $f^{(4)}_{5,6}(N)$2 and more generally for $f^{(4)}_{5,6}(N)$3.

Automated and Problem-Specific Probabilistic Constructions: Leveraging algorithmic probabilistic methods or even computational search could uncover configurations beyond the reach of classical random constructions.

Connections to Complexity Lower Bounds: Precise hypergraph Ramsey bounds, especially for explicit constructions, may inform lower-bound frameworks in complexity theory, particularly for threshold behaviors in monotone circuit complexity and property testing.

Conclusion

The paper provides a comprehensive solution to the Mubayi–Suk conjecture for generalized Erdős-Rogers functions with forbidden $f^{(4)}_{5,6}(N)$4 subhypergraphs in $f^{(4)}_{5,6}(N)$5-uniform hypergraphs, establishing $f^{(4)}_{5,6}(N)$6 behavior for $f^{(4)}_{5,6}(N)$7. The results extend to higher uniformities and yield improved lower bounds on off-diagonal Ramsey numbers, reflecting sophisticated use of the probabilistic method, extreme-case combinatorial analysis, and stepping-up lemmas. These advances deepen the understanding of induced subhypergraph avoidance phenomena and suggest concrete pathways for resolving the outstanding challenges in hypergraph Ramsey theory.